Finite Group DatabasebetaFinite Group Databasebeta| order | 8576 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C67 ⋊ #1289 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C67⋊#95 |
| commutator | C2×C4×C67 | C42 |
| Frattini | C23×C4 | C2×D67 |
| Fitting | #380×C67 | C2 |
| socle | C22×C67 | #95 |
| hypercenter | #380 | D67 |
| hypocenter | C67 | #1289 |
| LCS | G ▷ C2×C4×C67 ▷ C2×C67 ▷ C67 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #380 | |
228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,67|a^2=d,b^2=e,c^2=fg,e^2=fg,b^a=bc,c^a=cf,e^a=efg,d^b=dg,h^b=h^66956 conjugacy classes — view the full character table.
| maximal subgroup of | #1289×D67, C2×#22558076, #30745704, … |
| maximal normal subgroup of | #1289×D67, C2×#22558076, #30745704, … |
| maximal quotient of | #30742368, #30742379, #30742381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8576, 2377);